#7630863

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(5,6,8,10)(7,9,11,12)(13,14,20,17)(15,21,27,24)(16,25,23,19)(18,22,26,28);(1,2)(3,4)(13,15)(14,21)(16,28)(17,24)(18,25)(19,26)(20,27)(22,23);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,19)(15,22)(17,25)(18,21)(20,23)(24,26)(27,28);(13,17,20,14)(15,24,27,21)(16,25,23,19)(18,22,26,28);(13,18,20,26)(14,22,17,28)(15,25,27,19)(16,21,23,24);(5,7,8,11)(6,9,10,12)(13,19,20,25)(14,23,17,16)(15,26,27,18)(21,22,24,28)

Polycyclic presentation:

2^9|a^2=gh,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538059);